Surface dielectric response: Exact solution in the semiclassical infinite-barrier model with diffuse scattering

The dielectric response of a semi-infinite medium with a sharp electron profile at the surface is reexamined within the framework of the diffuse scattering model. The problem is shown to have an analytical solution for an arbitrary wave vector and frequency dependent bulk dielectric function (Formula presented). The explicit surface energy-loss function and the surface plasmon dispersion relation are derived, which extend the conventional ones (Formula presented) and (Formula presented), respectively(Formula presented) to the spatially dispersive case. These solutions are applied to the models with hydrodynamic, Lindhard, and Lindhard-Mermin dielectric functions. The results obtained give the deeper insight into the analytical structure of the surface dielectric response, in particular, showing that the coupling of the bulk plasmon with a charge outside a solid corresponds not to a pole, as is the case for a charge inside a solid, but to the branching point singularity in the energy-loss function. Comparison with Ritchie-Marusak theory of specular reflection shows that while retaining all the advantages of an analytical solution, the diffuse scattering model yields the more realistic description of the dynamical response of metal surfaces. The results are illustrated by the application to the medium with aluminum parameters and discussed in conjunction with charged particles energy losses.